Natural filtration of Schur functor

Let $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $$ be a short exact sequence of vector bundles. It is well known that for exterior powers there exists a natural filtration $$ 0=F_{r+1}\subset F_r\subset \cdots \subset F_0=\bigwedge^r B $$ such that $$ F_i/F_{i+1}\cong \bigwedge^i A\otimes \bigwedge^{r-i}C. $$

My question is whether an analogous statement holds for an arbitrary Schur functor. More precisely, given a partition $\lambda$, does $S_\lambda(B)$ admit a natural filtration $$ 0=F_N\subset F_{N-1}\subset \cdots \subset F_0=S_\lambda(B) $$ whose successive quotients satisfy $$ F_i/F_{i+1}\cong S_{\mu_i}(A)\otimes S_{\nu_i}(C), $$ and more generally whose associated graded object is $$ \operatorname{gr}\bigl(S_\lambda(B)\bigr) \cong \bigoplus_{\mu,\nu} \left(S_\mu(A)\otimes S_\nu(C)\right)^{\oplus c_{\mu,\nu}^{\lambda}}, $$ where $c_{\mu,\nu}^{\lambda}$ are the Littlewood--Richardson coefficients?

If so, I would appreciate a reference and, if possible, an explanation of how this filtration is constructed.